CRUSHING AND OEDOMETRIC DEFORMATION OF ROCKFILL …

Congresso de Metodos Numericos em Engenharia 2015Lisboa, 29 de Junho a 2 de Julho 2015

c©APMTAC, Portugal 2015

CRUSHING AND OEDOMETRIC DEFORMATION OFROCKFILL USING DEM

Joao Manso1∗, Joao Marcelino2 and Laura Caldeira2

1: Geotechnical DepartmentLNEC

Av. do Brasil 101, Lisbone-mail: [email protected], web: http://www.lnec.pt

2: Geotechnical DepartmentLNEC

Av. do Brasil 101, Lisbone-mail: [email protected],[email protected], web: http://www.lnec.pt

Keywords: Rockfill, DEM, particle breakage, crushing strength

Abstract. The behavior of rockfill may be simulated using the discrete element (par-ticle) modelling. Although its main advantage consists of not requiring the formulationof complex constitutive models, it requires extensive calibration to determine the particle–contact parameters in order to predict the macro-scale response. In this paper, the authorsperformed computer simulations of crushable agglomerates using distinct element method(DEM). These agglomerates were made by bonding elementary spheres using a stamp logic,in order to provide a statistical variability to the strength and shape of the agglomerates,similar to that of a real rockfill. This method was able to match the Weibull statisticsof the crushing strength of real rockfill grains, when tested individually between parallelplatens. Then, oedometric tests were simulated and compared quantitatively with the test-ing data. Several aspects were discussed, particularly, the effect of the loading rate on theposition of the compression curves regarding the practical question of performing DEMsimulations as fast as possible without creating inertia errors. They are also discussed inrelation to characterising crushable rockfill during oedometric compression. These sim-ulations of crushable agglomerates using DEM provides valuable insights concerning themicromechanical origins of rockfill compressibility. Once the model was calibrated throughthe results of tests performed, it can be used as a virtual laboratory to explore severalaspects, as size and shape of the particle’s sample. The model was capable of reproducingshort-term compressibility (oedometer) tests, but the simulation of long-term compress-ibility and creep or secondary coefficient are still under development. The model alsoprovided information on the evolution of the grain size distribution during loading of thespecimen.

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Joao Manso, Joao Marcelino and Laura Caldeira

1 INTRODUCTION

Rockfill material is widely used in many geotechnical engineering applications, such asrockfill dams and railroads, airports and railway embankments. Particularly, rockfill damshave been increasingly used due to their inherent flexibility, capacity to absorb largeseismic energy, and adaptability to various foundation conditions. They also became aneconomical option since the increasing use of modern earth and rock moving equipmentand locally available materials.Several researchers [1–7] defined rockfill material as being composed of more than 50%coarse-grained soil larger than the No. 4 sieve size [8]. The Portuguese Guidelines forDam Design [9] defined the upper limit of the particles size by construction issues, whereasthe lower limit could be the size of clay. Nonetheless, the behaviour of compacted rockfillshould not be influenced by the presence of smaller particles (less than 0,074 mm) andpermeability should be greater than 10−5 m/s . By definition in [10], rockfill used in roadembankments, should not have neither more than 30 % of material passing in ASTM sieve1”, nor more than 12 % of material passing ASTM sieve #200. Beyond that, the upperlimit of the particle dimensions should be 2/3 of the height of each construction layer or0,80 m. Therefore, within embankments, particles may cover a wide range of dimensions,from clayey materials to rockfill. In order to differentiate them, three categories areusually considered (Figure 1): soils, soil-rock mixtures and rockfill. In contrast to sandand gravel, rockfill grains are generally characterised by a lower crushing strength andexperience higher contact forces due to their larger size and coarse gradation.

Figure 1: Typical grading curve for: a – soil, b – soil-rockfill mixtures and c – rockfill.Adapted from [11]

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For a crushable rockfill material, particle breakage considerably influences its mechanicalbehaviour [1, 12–17]. Under compression or shear the crushing of particles modifies thegrain size distribution, increasing the percentage of fine material [18]. Consequently, thesedifferences in grain size distribution and in the available range of packing densities modifiesthe material permeability, its frictional properties and the critical states. Many researchers[12, 14, 15, 19, 20] experimentally investigated the role of particle breakage (large-scaletriaxial tests, multiaxial tests, and true triaxial tests) in the mechanical behaviour of thistype of geomaterials, characterising the influence of particle breakage in the strength ofrockfill materials [1, 13, 19, 21–23]. In order to comprehensively represent the degree ofparticle breakage during loadings, several researchers proposed different breakage indexes,based on particle-size distributions before and after a test [1, 12, 14, 17, 24]. Miura et al.(2003) [25] defined the breakage factor during the consolidation and shearing processusing increments of fines content (75 mm or smaller). Miura and O-Hara (1979) [26]considered the increase in the particle surface area to quantify the degree of particlecrushing in the loading process. On the other hand, Einav (2007) [27] included a relativeparticle breakage index based on fractal theory and, relying in this work, Wood and Maeda(2008) [28] proposed a grading state index to quantify the degree of particle breakage.Consequently, some researchers developed elastoplastic models [29, 30], bounding surfacemodels [31], and disturbed state concept models [19,22] incorporating the breakage indexin order to capture the stress-strain behaviours of crushable geomaterials.Rockfill materials particle sizes range from a few millimetres to over a metre, commonlyleading in rockfill dams to D50 sizes in the range 10–40 cm. Due to the large size parti-cles of such granular aggregates, testing them under oedometric, direct shear or triaxialconditions would require equipment of impressive dimensions. The largest testing cellsdescribed in the literature [1, 32–35] were only capable of testing rockfill aggregates withmaximum particle sizes not exceeding 15–20 cm [36]. A solution for this problem consistsof reducing the size of the rockfill materials for testing using modelling techniques [19] (Fig-ure 2a), such as: the scalping technique [37], parallel gradation technique [38], generationof quadratic grain-size distribution curve [39], and replacement technique [40]. Amongthese, Ramamurthy and Gupta (1986) [41] considered the parallel gradation method asthe most appropriate. However, rockfill behaviour depends on its grain size scale, as shownby tests performed on materials with different mean grain size, but equivalent grain sizedistributions [1,32,33,36,42]. Particle breakage controls rockfill mechanical behaviour andis affected by scale effects, then, it is expected that scale effects are present in the con-stitutive behaviour of rockfill [36]. As it is almost impossible to test real size specimens,researchers studied alternative procedures to extrapolate results from a reduced scale toin situ conditions. Although it is useful to propose rules to scale some rockfill properties,such as strength [43], difficulties arise when considering other aspects of constitutive be-haviour [36]. The discrete element method [44] can be an effective tool for investigatingsize effects, provided it is capable of properly simulate grain failure mechanisms for severalstress paths [17,45–51].

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Figure 2: Test grading curves

Several approaches were adopted to study particle breakage using DEM. Some consid-ered subparticles joined by bonding or cohesive forces [45,52,53,53–56], another approachreplaced a particle which verified a predefined failure criterion with an equivalent groupof smaller particles [57–60]. These techniques were employed with either discs in 2D orspheres in 3D. When considering particles with general shapes, the technique consistedof bonding unbreakable and nondeformable subparticles creating a breakable particle.Then, if the bond between these subparticles broke, breakage occurred [61,62]. Other re-searchers proposed a method that combined finite element method (FEM ) with discreteelement implementations [20, 63]. In this method, particle movements and interactions

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were determined using DEM ( [64]) and the deformation of the rockfill material wassolved computationally according to the rheological behaviour of the material, adoptinga finite element mesh for each particle. A natural evolution of the DEM is to be capa-ble of reproducing the complex mechanical behaviour of granular geomaterials, such asdeformability and crushability, and to simulate accurately and conveniently the particlebreakage phenomenon, despite the complexity of the considered particle shapes.This paper deals with laboratory testing and discrete element modelling of rockfill mate-rials collected from a dam site located in the north of Portugal. The model parameterswere calibrated through results of tests performed on samples with a given grain size dis-tribution. Then, the model allowed to study the influence of density and pressure on thestrength and deformation of rockfill material through the loading steps of an oedometriccompression test. This model and the basic mechanical behaviour of the material de-rived from Montesinho dam would be the same as that of rockfill materials at other sites.However, the strength and deformation of rockfill materials depend on the rock type andmineralogy and some of the material constants may be different for rockfill materials atother sites.

2 MODELLING OF A ROCKFILL PARTICLE

2.1 Contact constitutive models

The PFC2D program [65] uses the distinct element method with the soft contact approach.This approach assumes that elements have a finite normal stiffness and allows bodies tooverlap in order to represent elastic flattening at contacts. The constitutive representationof contact points between two elementary spheres include a stiffness model, a bondingmodel and a slip model. In the linear contact-stiffness model, the normal and sheartangent stiffness at a contact, Kn and Ks respectively, are computing assuming thatthose stiffnesses act in series, which can be written as:

Kn =kAn k

Bn

kAn + kBn(1)

and

Ks =kAs k

Bs

kAs + kBs(2)

where kAn and kBn represent the normal stiffness of two contacting objects A and B, andkAs and kBs represent the shear stiffnesses, expressed as the force per unit displacement.The bonding model resembles a pair of elastic springs at a point of glue. It limits bothnormal and shear forces that a contact can carry by imposing bond-strength limits. Whena bond is created between two spheres, the maximum shear force that it can sustain intension is specified as well as the the maximum shear force it can bear before breaking.Note that these values may be modified at any time during the simulation. When one

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of these values is exceeded, the bond breaks. This type of bond does not resist to abending moment, due to the fact that it acts over a considerable small area of contactpoint. Therefore, if a third sphere does not exist to restrain the motion, it has no rollingresistance [45].Finally, the slip model acts between two unbonded spheres in contact or after a bondbreaks between two bonded spheres. It allows the contacting spheres to slip at a limitingshear force, governed by Coulomb’s equation, and limits the shear force between them.When two spheres have different values of friction coefficients, the smaller one is used tocalculate the maximum allowable shear force before sliding occurs:

F smax = µ|F n

i | (3)

where F smax represents the maximum shear contact force, where µ is the coefficient of

friction, and F ni represents the normal component of the contact force.

2.2 Variability implementation

The agglomerates intended to model rockfill particles should be capable of simulate thecrushing strength of a real grain, due to diametral breakage of the bonded agglomeratesbetween flat platens, and at the same time reproduce realistic Weibull distributions ofcrushing strength for a group of specimens. In order to achieve this, some considerationshad to be made. Following [45], to avoid the effect of locked-in forces, that emerge fromoverlaps between spheres when creating a bond (which would release strain energy if thebond is broken), a sphere should be replaced by a group of agglomerates made from aregular assembly of spheres in hexagonal close packing (HCP), without initial overlap.The main purpose of this regular packing is to minimise the space between the spheres ofthe agglomerate [45,66]. Then, in order to provide a statistical variability to the strengthand shape of the agglomerates, similar to that of a real rockfill, we introduced the stamplogic presented by [67], which is illustrated in Figure 3.This logic creates a clump by stamping a circled area that corresponds to the desiredgrain size. The centre position of the particles that lie within this area are added to aclump and particles grouped in this way intend to represent a grain acting as a singleparticle. The clump size is defined by specifying the radius of the stamp circle with astandard deviation and clump stamping is continuously activated until all particles in theassembly belong to a clump (Figure 4b).

3 MODELLING OEDOMETRIC COMPRESSION OF A ROCKFILL SYS-TEM

Each numerical test performed in this work started by create an initial set of exo-spheres,which consisted in placing them at random with a size slightly smaller than the required,without overlaps between themselves. Then, they were expanded to the final size andcycled until equilibrium was reached, reducing unwanted gaps. Following [45, 66] shear

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and effective for modeling of rock like brittle material inPFC. In this section we apply the clump technique tosimulate Lac Du Bonnet granite and compare thesimulated results with laboratory test results.

The clump logic introduced in Fig. 15 uses maximumclump size to form irregular grain modeling by limiting thenumber of particles in one clump. This type of size controlis, however, not adequate to consider the actual grain sizesince it only defines the maximum number of particles so itis difficult to account for the actual grain size in real rock.For this reason, we introduced the stamp logic for the sizecontrol of the clump. This logic is illustrated in Fig. 16Using this logic, a clump can be created by stamping acircled area that corresponds to the desired grain size sothat the particles within this area, if their center position isinside the stamped boundary, they can be added to a clumpand grouped particles in a clump represent a grain acting asa single particle. The size of a clump is determined byspecifying the radius of the stamp circle with a standarddeviation (i.e. ‘‘clp_rad’’ in Table 3) and clump stamping iscontinuously activated until the 99% of the particles in theassembly are clumped.

While we were not intending to match the similarity ingrain size with actual grains, the final average clump sizecalibrated for the simulation was 2mm in diameter which isthe same average grain size as Lac du Bonnet granitereported by Martin et al. [11]. However, no furtherattempts were made to match the similarity in statisticaldistribution and the shape of grains to actual grains.

As discussed earlier, the stiffness ratio (i.e. normal toshear ratio for both contact and bond) directly involvesPoisson’s ratio. However, as Potyondy and Cundall [1]mentioned, direct comparison of Poisson’s ratio in PFC2D

with laboratory result is not that meaningful because of thelimitation in 2D analysis of PFC. The fundamentalassumption for particle elements employed in PFC2D isthat particles are either disks having finite thickness orspheres having single layer toward out of plane direction.The former is similar to the plane strain condition and the

latter is similar to the plane stress condition in continuummechanics. However, unlike continuum mechanics assump-tion, there is no out of plane stress in plane strain and noout of plane strain in plane stress in PFC2D. Hence,attempts to match volumetric strain or Poisson’s ratio inPFC2D material to match 3D physical material are notalways successful [1].Hence, no attempt was made for calibrating the ratio

with laboratory results and only to be fixed to 2.5 which isidentical value that Diederich [2] provided using simplebonded particle assembly (i.e. no clump or no clusterparticle assembly) for the same rock.The bond normal strength (sn) was set to one and half to

two times of actual tensile strength of Lac Du Bonnetgranite. Uniaxial strength of this synthetic assembly wasthen calibrated by increasing clump size (clp_rad) and shearto normal strength ratio ss=sn

� �. Both parameters could

increase both uniaxial strength and slope of failureenvelope. As discussed earlier, increasing ss=sn ratioimplies that as the micro-shear strength is set to highervalues than that of normal strength only tension cracks canoccur. Increasing clump size has largely two effects on thestrength. First, once particles are joined to the clump, theindividual particle rotation is suddenly frozen, so if clumpradius is larger and larger, then particle rotation is moreand more suppressed and the strength will increase (i.e.note that suppressing particle rotation highly increasesmaterial strength). Secondly, increasing clump radius cangeometrically increase the apparent friction along theclump boundaries, thus this friction could effect bothuniaxial strength and slope of failure envelope.Fig. 17(a) and (b) show the comparison of axial

stress–strain and lateral strain response, respectively,between PFC synthetic rock and Lac Du Bonnet granitefor 15MPa of confining stress. Although PFC exhibits aslightly higher strength and lateral strain shows somewhathigh dilation, overall responses for axial stress–strain arewell matched. The stress–strain in PFC is somewhatdifferent from the laboratory results after the post-peakbecause the servo control of strain in laboratory test setupwas not reflected in current PFC simulation. WhileHazzard and Young [10] demonstrated that post-peakresponse of Westerly granite could be captured in PFCusing constant acoustic emission rate since our study wasnot focused on the post-peak response, this issue was notaddressed in our current study.

ARTICLE IN PRESS

Clump

Stamp radius

1.0 0.2mm

Clump Stamp

Fig. 16. Illustration of stamping logic to control clump size.

Table 3

Micro-parameters of Lac du Bonnet granite used for calibration

Rmin 0.20mm Ec 20.0GPa

Rmax=Rmin 1.5 kn=ks 2.5

l 1.0 Ec 20.0GPa

m 0.1 kn=ks 2.5

sn 25.0�3.5MPa clp_rad 1.070.2mm

ss=sn 20.0 Unit_weight 2639kg/m3

N. Cho et al. / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 997–10101006

Figure 3: Illustration of stamping logic to control clump size [67]

(a) Unclumped particle(b) Clumped particle

Figure 4: Introducing variability in a typical rockfill particle

stiffness and friction were reduced to zero, while lateral wall stiffness was reduced ten timesthe initial normal stiffness, which was increased 100-fold during the process of preparingthe sample (Table 1). In order to insert the aggregates a linked list storing the coordinatesof exo-spheres centres was created and then they were deleted. In their place, with thecentre located at the list coordinates, were created aggregates, and applied the clump logicuntil all particles were clumped. After this, the new assembly was cycled until equilibriumwas reached again, before starting the test. At this time, the shear and normal stiffnessesof the balls of the agglomerates were set to their final value (3 · 109 N/m).In order to reduce the possibility of bonds breaking during the preparation of the specimen,

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Input parameter Numerical valueFriction coefficient 0

Shear stiffness [N/m] 0Normal stiffness [N/m] 5 · 1010

Wall stiffness [N/m] 5 · 107

Table 1: Adopted properties for specimen preparation

their strengths were initially set very high (1 ·109 Pa) until equilibrium was reached. Thisobjective was achieved and no bonds broke during the sample preparation process, afterintroducing the clumps, for an equilibrium stress of 5 kPa. Finally, before starting thetest, bond strengths were fixed to their final value (1 · 106 Pa) and friction coefficient wasset to 0.5, corresponding to a contact friction angle of 26.5 (Table 2).

Input parameter Numerical valueFriction coefficient 0.5

Normal and shear stiffness [N/m] 3 · 109

Normal and shear stiffness (parallel bonds) [N/m] 3 · 109

Normal and shear strength (parallel bonds) [MPa] 1 · 106

Table 2: Adopted properties for test modelling

Figure 5 shows the layout of particles used in this work. The characteristics of the modelare presented in Table 3, which resulted in 1520 agglomerates. The model was thenuniaxially compressed, in 100 kPa stages (until 1 MPa), by moving the top and bottomwalls progressively together and fixing the position of the other pair of walls, to achievethe desired stress path. Following [45], the walls moved at a maximum speed of 0.01 m/swhich were controlled by a servomechanism to reach the desired stress. This maximumspeed was considered slow enough to eliminate rate effects, due to any bouncing that couldoccur initially for unloaded agglomerates. The stress was determined by summing andaveraging all contact forces on the top wall. The voids ratio was calculated consideringthe solids volume as the total area of the circles, resulting in an initial value of 0.25.Several velocity-limited loading tests were performed in order to check for possible inertiaeffects on the location of the virgin compression line. The limited velocities started witha value of 2 m/s and decreased until 0.01 m/s. It was considered that no dynamic effectoccurred on the compression behaviour for platen velocities below 0.01 m/s (Figure 6b).However, when the velocities were allowed to reach higher values, an increase of thestrength of individual agglomerates was registered, as seen in Figure 6a, which meantthat grains would start crushing at a higher stress level. Furthermore, when analysingthe numerical timestep at which bond breakage happened, they appeared for a smaller

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Figure 5: Rockfill specimen of 1520 agglomerates

Input parameter Numerical valueSample height [m] 0.474Sample width [m] 0.500

Rmin [mm] 0.0095Rmax [mm] 0.0191

Clump size [mm] 1.0± 0.2Density [kg/m3] 2052

Table 3: Agglomerate characteristics adopted to model a typical rockfillparticles

timestep in the case of higher compression rates (Figure 7). The resulting behaviour was,as observed in Figure 6b, equivalent, leading to a similar amount of bond breakage.Since the beginning of the oedometric test irrecoverable compression was registered, whichhappened beyond 100 kPa. This occurred without any significant breakage, so this be-haviour can be attributed to agglomerates rearrangement due to elastic compression.The breakage started at approximately 400 kPa (for a limiting velocity of 0.01 m/s) and600 kPa (for a limiting velocity of 0.5 up to 2 m/s).

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Joao Manso, Joao Marcelino and Laura CaldeiraNumber of bonds

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Figure 6: Effect of limiting compression rate on oedometric compression curve

Nakata et al. (2001) [68] compared the results of single-particle crushing tests with theone-dimensional compression of samples of the same uniformly graded sand. The resultsshowed that the macroscopic stress level required to cause crushing and irrecoverablecompression in a sand sample, was much smaller than that required to break individualgrains. They attributed this to the unequal distribution of internal contact forces within

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Joao Manso, Joao Marcelino and Laura CaldeiraNumber of bonds

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Figure 7: Numerical timesteps against bond breakage

an aggregate of grains. Cundall and Strack (1979) [44] found that only a number ofheavily loaded chains of particles respond to an externally applied stress, which hadbeen observed experimentally by Oda and Konishi (1974) [69]. Whereas, the remainingparticles within the mass are only slightly loaded [70]. Their main contribution to thesystem is the stabilisation of the main loading chains. As a result of this particle breakage,several authors [71–74] have studied diametrically loaded single particles. Cheng et al.(2001) [71] described two breakage phases: an initial one, governing the beginning ofindividual grains breakage, and a second one in which further breakage continued asthe compression proceeded. In a rockfill sample, agglomerates are supported by nearbyagglomerates, increasing its coordination number. This aspect should distribute stressesalong its contacts, leading to a reduction in tensile stress, when compared to single particlecrushing tests [45, 75]. Therefore, an agglomerate in a rockfill sample might break at anapplied stress even higher than the crushing strength.As in the work of Cheng et al. (2003) [45] regarding a soil sample, there is an interestingcoincidence between the beginning of linear bond breakage plot against the logarithm ofmean stress and when rockfill reaches the linear logarithmic compression line at approxi-mately 600 kPa, which is clearer for a limited velocity of 0.01 m/s. This macro behaviourof the simulated rockfill can be referred as clastic yielding and happens when the appliedstress causes the onset of particle crushing, assuming that the onset of particle breakageleads to the bend of the normal compression line, causing the rapid increase of the ma-terial compressibility index [76]. During unloading (swelling curve) the simulated rockfillelement showed an elastic behaviour and no bond breakage occurred.Figure 8 compares isotropic compression curves between the rockfill sample and the DEM

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simulation with a platen speed of 0.01 m/s. Both shapes are similar and the DEMsimulation captures the transition from particle rearrangement, due to elastic compression,into what may be described as clastic compression. We are aware that due to the natureof the real material, there are greater variability in the real rockfill. Comparing rockfillparticles with the agglomerates in the DEM simulation, it is clear that there is a greatervariety of sizes and asperities in the real material and that agglomerates have only a limitednumber of component spheres [45]. These particularities may explain the differencesbetween the isotropic compression line, specially concerning creep that appears in thematerial at the end every load step and it is not reflected numerically (Figure 8 and9). The rockfill sample continues fragmenting at constant loads, whereas the simulatedrockfill remains stable when the increase of load stops. This can be also attributed to thelinear contact-stiffness adopted and could be a matter of selecting a different one thatconsidered decrease of strength with time. Some researchers [77] suggested that in orderto improve numerical results, the agglomerates should be modelled using spheres withdimensions close to the comminution limit of silica sands, but this has for now unrealisticcomputation costs. Sheet2

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Figure 8: Normalised oedometric compression curves

4 CONCLUSIONS

Short-term compressibility on rockfill behaviour (oedometer tests) was successfully ap-proached through DEM modelling provided a careful specimen preparation in the modeland defining the parameters based on laboratory test results.

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0 20 40 60 80 100 120

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Figure 9: Oedometric test stress strain curves

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The developed model adopted the clump logic to simulate rockfill particles allowing themto randomly subdivide into smaller shapes. This has been achieved by introducing the ideathat a rockfill grain can be considered to be an agglomerate of bonded micro-elements,here represented as clumps. Breakage occurred in the specimen when a bond betweensubparticles broke. This approach seemed to be capable of creating a lifelike distributionof grain crushing strengths and rockfill compressibility behaviour.Several test simulations were conducted limiting wall velocities up to 2 m/s in order toperform DEM simulations as fast as possible without creating inertia errors.In further developments, the simulation of long-term compressibility and creep or sec-ondary coefficient may be achieved by including a particle breakage criterion, which in-corporates the mechanics of crack propagation in brittle materials. Furthermore, modelswill be developed in order to simulate triaxial tests with different stress paths and becapable of replicating the mechanical behaviour of rockfill, particularly its strength, dila-tancy, and critical states. Special attention will be given to the evolution of the grain sizedistribution during loading of specimens for a wide range of particle sizes. This study onthe micromechanics of granular interactions should lead to improved continuum rockfillmodels.

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